The problem of computing the exact stretch factor (i.e., the tight bound on the worst case stretch factor) of a Delaunay triangulation is one of the longstanding open problems in computational geometry. Over the years, a series of upper and lower bounds on the exact stretch factor have been obtained but the gap between them is still large. An alternative approach to solving the problem is to develop techniques for computing the exact stretch factor of ``easier'' types of Delaunay triangulations, in particular those defined using regular-polygons instead of a circle. Tight bounds exist for Delaunay triangulations defined using an equilateral triangle and a square. In this paper, we determine the exact stretch factor of Delaunay triangulations defined using a regular hexagon: It is 2. We think that the main contribution of this paper are the two techniques we have developed to compute tight upper bounds for the stretch factor of Hexagon-Delaunay triangulations.

中文翻译：

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Hexagon-Delaunay 三角剖分的拉伸因子

计算 Delaunay 三角剖分的精确拉伸因子（即最坏情况拉伸因子的紧界）的问题是计算几何中长期存在的开放问题之一。多年来，已经获得了一系列精确拉伸因子的上下界，但它们之间的差距仍然很大。解决该问题的另一种方法是开发用于计算“更简单”类型的 Delaunay 三角剖分的精确拉伸因子的技术，特别是那些使用正多边形而不是圆定义的三角剖分。使用等边三角形和正方形定义的 Delaunay 三角剖分存在紧边界。在本文中，我们确定使用正六边形定义的 Delaunay 三角剖分的精确拉伸因子：它是 2。